Surface Area of Cone

The surface area of a cone is the amount of area occupied by the surface of a cone. A cone is a 3-D shape that has a circular base. This means the circular base has a radius and diameter. The distance between the center of the base and the topmost part of the cone is the height of the cone. When we refer to the surface area of a cone, it usually refers to the total surface area. When we refer to the curved surface area of a cone, it means we referring to the lateral surface area of the cone.

In this article, we will learn how to calculate the surface area of a cone. The total surface area includes the curved and the flat circular area whereas the curved surface area includes the area of only the curved surface.

The area occupied by the surface/boundary of a cone is known as the surface area of a cone. It is always measured in square units. Stacking many triangles and rotating them around an axis gives the shape of a cone. Since it has a flat base, it has a total surface area as well as a curved surface area. We can classify a cone as a right circular cone or an oblique cone. The vertex in the right circular cone is usually vertically above the center of the base, whereas, the vertex of the cone in an oblique cone is not vertically above the center of the base.

Since a cone has a curved surface, we can express its curved surface area as well as the total surface area. Observe the cone given below to observe its surface area, the height of the cone (h), the slant height (l), and the radius (r).

A cone has two kinds of surface areas.

• Total Surface Area
• Curved Surface Area

If the radius of the base of the cone is 'r' and the slant height of the cone is 'l', then the surface area of a cone is calculated using 2 formulas:

• Total Surface Area, TSA of cone = πr(r + l)
• Curved Surface Area, CSA of cone = πrl

Sometimes, we express these formulas in terms of the height of the cone. Observe the cone given above to see that the height (h) of a cone is different from the slant height (l) of the cone. Therefore, using the Pythagoras theorem on the cone, we can find the relation between the surface area of the cone and its height. We know, h² + r² = l², where 'h' is the height of the cone, 'r' is the radius of the base, and 'l' is the slant height of the cone. So, using the Pythagoras theorem, we get, l = √(h² + r²)

Thus,

• The total surface area in terms of height is expressed as, TSA of cone = πr(r + l) ⇒ TSA of cone = πr[r + √(h² + r²)].
• The curved surface area of the cone in terms of height is expressed as CSA of cone = πrl ⇒ CSA of cone = πr[√(h² + r²)].

Like other three-dimensional shapes, a cone also has flat and curved surfaces. The curved surface area of cone refers only to the curved part of the cone which is other than the circular flat base. To find the curved surface area of a cone, we multiply the radius and slant height of the cone by pi(π). Let us derive the formula for the curved surface area of cone which is also referred as the CSA of cone. The Curved surface area is also known as the lateral surface area of a cone

Curved Surface Area of Cone Formula

The curved surface area of the cone can be given by finding the area of the sector by using the formula,

Area of the sector (in terms of length of arc) = (arc length × radius)/ 2 = ((2πr) × l)/2 = πrl

∴ The curved surface area of a cone, S = πrl

In order to derive the total surface area of cone formula, let us cut the cone open from the center which looks like a sector of a circle (a plane shape). Let us take a cone of height 'h', base radius 'r', and slant height 'l.

The total surface area of cone = base area of cone + curved surface area of a cone

⇒ Total surface area of cone = πr² + πrl = πr (r + l)

∴ The total surface area of cone, T = πr (r + l)

Total Surface Area of Cone

Let us find the total surface area of cone and the curved surface area of cone.

Example: Find the total surface area and curved surface area of the cone whose radius is 7 inches and slant height is 3 inches. (Use π = 22/7)

We know, the total surface area of the cone is πr (r + l), and the lateral surface area of a cone is πrl. Given that: r = 7 inches, l = 3 inches, and π = 22/7

Thus, total surface area of cone, T = πr (r + l) = (22/7) × 7 × (7 + 3) = (22/7) × 7 × 10 = 22 × 10 = 220 in²

∴ The total surface area of the cone is 220 in²

The curved surface area of the cone, S = πrl = (22/7) × 7 × 3 = 66 in². Therefore, the curved surface area of the cone is 66 in².

Let us look at some more examples of the surface area of a cone for a deeper understanding.

What is the Surface Area of a Cone?

The area occupied by the surface of a cone is referred to as the surface area of the cone. There are two types of surface areas of a cone, the total surface area and the curved surface area of the cone.

How to Find Surface Area of Cone?

The surface area of a cone can be found by using the following steps:

• Step 1: Identify the given values of radius, slant height, and height of the cone.
• Step 2: Use the appropriate formula to calculate the surface area. If the total surface area of the cone has to be found, use the formula, total surface area of the cone = πr (r + l), and if the curved surface area of the cone has to be found, use the formula, curved surface area = πrl
• Step 3: Simplify and write the answer in square units.

What is the Curved Surface Area of Cone?

The area of the curved surface of a cone is known as the curved surface area of the cone. The formula that is used to calculate the curved surface area of a cone is πrl, where 'r' is the radius of the base and 'l' is the slant height of the cone. In this, we do not consider the area of the base of the cone which is in the shape of a circle.

What is the Curved Surface Area of a Right Circular Cone?

A right circular cone is defined as the cone whose axis is the line joining the vertex and the midpoint of the circular base. Thus, the curved surface area of a right circular cone is given as πrl where 'r' is the radius of the base and 'l' is the slant height. In terms of height, the curved surface area of a right circular cone is given as πr[√(h² + r²)] where 'h' is the height of the right circular cone.

How to Find the Surface Area of a Cone with Slant Height and Diameter?

The total surface area of a cone with slant height and diameter of the cone can be found by using the formula, T = π(D/2) ((D/2) + l), where D is the diameter and l is the slant height. The curved surface area of the cone with slant height and diameter can be found by using the formula S = π(D/2)l, where D is the diameter and l is the slant height.

How to Find the Total Surface Area of a Cone?

The total surface area of a cone can be found by using the following steps:

• Step 1: Check the values given in the question.
• Step 2: Substitute the values of radius and slant height in the formula πr (r + l). In case, if there is no slant height given, we write the value of slant height in terms of the height of the cone, 'h', by substituting l = √(h² + r²) which gives the value, T = πr(r + √(h² + r²)) where 'r' is the radius of the cone and 'h' is the height of the cone.
• Step 3: Now, find the value of the total surface area.
• Step 4: Write the final answer in terms of square units.

How to Find the Curved Surface Area of a Cone?

The curved surface area of a cone can be found by using the steps given below:

• Step 1: Write the values given in the question.
• Step 2: Substitute the values of radius and slant height of the given question in the formula πrl. In case, if there is no slant height given, we write the value of slant height in terms of the height of the cone, 'h', by substituting l = √(h² + r²) which gives the value, S = πr√(h² + r²) where 'r' is the radius of the cone and 'h' is the height of the cone.
• Step 3: Now, find the value of the curved surface area and write the answer in terms of square units.

What is CSA of a Cone?

The CSA of a cone is the abbreviated form of the Curved Surface Area of cone. The formula that is used to calculate the curved surface area of a cone is, CSA of cone = πrl, where 'r' is the radius of the base and 'l' is the slant height of the cone. Here, we do not consider the area of the base of the cone which is in the shape of a circle.

What is TSA of a Cone?

The TSA of a cone is the abbreviated form of the Total Surface Area of a cone. The formula that is used to calculate the total surface area of a cone is, TSA of cone = πr(r + l), where the radius of the base of the cone is 'r' and the slant height of the cone is 'l'. Sometimes, the total surface area can be calculated in terms of the height of the cone. So, if 'h' is the height of the cone, r is the radius of the base, and l is the slant height of the cone, then the total surface area in terms of height can be given as, TSA of cone = πr[r + √(h² + r²)]. In this formula, 'l' is replaced by l = √(h² + r²)

What is Lateral Surface Area of a Cone?

The lateral surface area of a cone (LSA of cone) is also known as the curved surface area of a cone. The curved surface area of a cone refers only to the curved part of the cone which is other than the circular flat base. To find the curved surface area of a cone, we multiply the radius and slant height of the cone by pi(π), and the formula which is used to find the lateral surface area of a cone is, CSA of cone (LSA of cone) = πrl, where 'r' is the radius of the base and 'l' is the slant height of the cone.